A cylindrical can is to be made to hold 1000cm^3 of oil. How do you find the dimensions that will minimize the cost of metal to manufacture the can?

Question

Cameron Alexander · Accepted Answer

Height = 10.84 cm, radius of the base = 5.42 cm and material for the surface =37 square cm, nearly

Let height = h and radius of the base = r. Then, volume #V=pi r^2h=1000# cc and surface area of the can #S=2pir^2+2pirh# Now, eliminating h, #S=S(r)=2pi(r^2+1000/(pi r^2))# #S'=2pi(2r-1000/(pir^3))=0#, when #r=(1000/(2pi))^(1/3)=5.42# cm, nearly. Correspondingly, h= 10.84 cm, nearly There is no maximum for S. Also, #S''=2pi(2+3000/(pir^4))>0#.. For this r= 5.42 cm, S = 37.1 #cm^2#

Christopher Allers · Answer

undefined To minimize the cost of metal to manufacture the cylindrical can holding 1000 cm^3 of oil, you would need to find the dimensions that minimize the surface area of the can. The formula for the surface area of a cylinder is given by A = 2πrh + πr^2, where r is the radius and h is the height. Using the volume formula V = πr^2h, you can express the height h in terms of the volume V and the radius r. Then, substitute this expression for h into the formula for surface area A. Finally, differentiate the resulting expression for A with respect to r, set it equal to zero, and solve for r to find the radius that minimizes the surface area, hence minimizing the cost of metal.